Relation to the perturbation method

An alternative approach for the examination of the effect of the pipe geometry on the eigenfre-quencies of the pipe can be developed usingperturbation theory. This approach has already been used by several authors for the investigation ofvocal tractshapes. Schroeder [130] developed a perturbation method for vocal tract cross sectional area changes based on the Ehrenfest theo-rem [54]. Later, Story [134] has applied this technique successfully for “tuning” the area function

of vocal tracts. He sensitivity

functions proposed an algorithm based on so-called sensitivity functionsto match the

natural resonance frequencies of the vocal tract to predefined values by changing the cross sec-tional area of the tract. This approach was extended by Adachiet al.[9] for the perturbation of the length of the tract applied for male–female vocal tract shape conversion. In the following the sensitivity-function-based technique is compared to the optimization algorithms introduced above.

To establish the sensitivity functions theEhrenfest theoremis used. The theorem states that if an oscillation system is excited in itsnth eigenmode with the frequencyfnand energyEn, then the ratioEn/fnremains unchanged as the system is perturbed in an “adiabatic” manner [54]. In case of organ pipes (or vocal tracts) “adiabatic” means that a large number of oscillation periods pass during the perturbation [9]. Thus, the change of the natural frequencyδfncan be calculated from the change of energyδEnas

Ehrenfest theorem δfn

f_n =δEn

E_n . (5.11)

To evaluate the change of energy δEn the so-called acoustic radiation forceis used [23]. By means of the radiation force the work done by the adiabatic change is expressed and hence the change of energy in the systemδEn can be determined. The radiation forceP(z)for an imper-meable wall is given as [130]

radiation force

P_wall(z) = PE(z)−KE(z), (5.12)

Figure 5.4. Optimization using the cost function. Comparison of the original and optimized input admit-tances (above). Modal waveforms corresponding to the eigenfrequencies of the optimized pipe (below).

wherePEandKEdenote the potential and kinetic energy densities, respectively, which are ob-tained as¹

PE(z) =1 2

1 ρ0c²

p²(z, t)

(5.13a) KE(z) =1

2ρ0

U²(z, t) S²(z)

, (5.13b)

withS(z)representing the area andh·idenoting temporal averaging. For an opening located at the coordinatezethe radiation force is attained as [9]

P_exit= TE(z_e) = PE(z_e) + KE(z_e), (5.14) whereTEis the total energy density.

The change of energy can be expressed from the radiation force. Here, we limit ourselves to the case of concatenated cylindrical ducts, such as chimney pipes. A more general discussion is found in [9, 130]. It is assumed that the geometry is composed ofNscylindrical ducts, with area Si, lengthLi, occupying the regionsz_i−1 ≤z ≤zi. i = 1,2. . . Ns. The perturbation consists of changes of two kinds: (1) the change of the cross sectional areas, represented byδSi, and (2) the change of the lengths of the sections, denoted byδLi. The changes of energy due to area and length perturbations are found following [9] as

δE_n^(S)=

Ns

X

i=1

δE_n,i^(S)=−

Ns

X

i=1

δS_i Z z_i

zi−1

P_n(z) dz, (5.15a)

δE_n^(L)=

N_s

X

i=1

δE_n,i^(L)=−

N_s

X

i=1

δLiSiTEn,i. (5.15b)

1For the evaluation of the potential and kinetic energies the nonlinear Euler equation (3.9) must be dealt with. The derivation of equations (5.13) is found in [9].

5.2. METHODOLOGY 59 The upper indices (S) and (L) refer to changes due to area and length perturbations, respectively.

The notationsPnandTEndenote that the radiation force and the total energy density are eval-uated for the nth natural resonance. Since the total energy densityTEn,i is constant in each cylindrical section of the pipe, the integration boiled down to a multiplcation in (5.15b).

To apply the perturbation method for the optimization of the geometry it is useful to express the sensitivity functions for the area and length changes. The sensitivity functionsG^(S)_n,i andG^(L)_n,i relate the relative change of the area and the length of theith section, respectively, to the relative change of thenth natural resonance frequency. Using (5.15) we get

length and By means of the d’Alembert solution (5.4) the radiation force and the potential, kinetic, and total energies can be expressed and their integral for each cylindrical section can be evaluated analyt-ically. Thus, the sensitivity functions in (5.16) are also obtained in an analytical manner.

An automatic optimization procedure for modifying the tract area function was introduced by Story [134] and has been extended for length modifications by Adachiet al.[9]. Here, the latter iterative algorithm is discussed briefly. Let us denote the normalized difference between the actual resonant frequency and the target frequency byζn, using the same notations as above

ζn= nf1−f_n^∗

f_n^∗ . (5.17)

The iterative update rule introduced by Adachiet al.[9] reads as

iterative HereNfdenotes the number of eigenfrequencies to be tuned, the upper indices(m)and(m+ 1) refer to the original and updated values in themth iteration, respectively. The parametersαand βcontrol the perturbation amplitudes of the area and the length, respectively, and are determined in an experimental manner to provide convergence. The algorithm of Adachiet al.implements constraints for the maximal length and the minimal area, and a limit that prevents the solution from becoming unnaturally discontinuous [9]. These constraints are not considered here.

In order to test the applicability of the sensitivity approach for chimney pipe optimization the iterative algorithm introduced above was implemented. In the following we focus on the sim-plest case of optimizing the length of the main resonator and the chimney, as already introduced in Section 5.2.2. In this case only (5.19) is used in the iterative algorithm. Starting from the given initial main resonator and chimney lengths, the sensitivity functions (5.16) are evaluated. The normalized differencesζn are obtained using (5.17) by evaluating the input admittance function and finding its corresponding peaks. Then, either the chimney or the main resonator length is updated, in an alternating manner, using the update rule given in (5.19). The same steps are re-peated untilζnare all sufficiently small, or the maximum number of iterations is reached without convergence.

The results of the sensitivity-function-based iterative algorithm are discussed in the follow-ing for the pipe introduced above (D_P = 79.00 mm,D_C = 28.72 mm, W_M = 59.99 mm,H_M =

Figure 5.5.Convergence of the sensitivity-function-based approach for chimney pipe optimization

25.66 mm, andT0= 20^◦C). The goal fundamental frequency wasf1= 140.0 Hzand the fifth har-monic partial was selected for amplification. Unlike in the case of the heuristic iterative algorithm and the cost function method discussed in the previous sections, it should also be determineda prioriwhich eigenfrequency will amplify the selected partial to be able to use the sensitivity func-tions. Here, based on the results shown in Figure 5.4, the fourth eigenfrequency was selected.

The critera of convergence were chosen as that both the first and the fourth eigenfrequency must differ lest than 25cents (< 1.5%) from the target frequencies. The maximal number of itera-tions was set as25. Among a few values triedβ = 1seemed a reasonable choice providing fast convergence in a wide range of initial lengths.

Figure 5.5 shows the results of the optimization based on sensitivity functions for different initial main resonator and chimney lengths. The displayed residual is the maximum of the nor-malized errors, i.e.e_res = max{ζ₁, ζ₄}. As shown by the white area in the diagram, the algo-rithm provides convergence in a wide range of initial resonator and chimney lengths. It was found that convergence is obtained in less than ten iterations in the whole region of conver-gence. Depending on the starting point of the algorithm, different solutions are reached with resulting main resonator and chimney lengths in between 531.9 mm ≤ L_P ≤ 600.5 mm and 238.5 mm ≤ L_C ≤ 448.6 mm, respectively; with longer chimneys going with shorter main res-onators andvice versa.

Interestingly, the optimum point found by the heuristic iterative algorithm (marked by the black diamond marker in Figure 5.5) is not near the center of the region of convergence, as far as the initial length of the chimney is considered. When the initial chimney length is much greater than the optimal, convergence is still reached; however, if the initial chimney length is decreased a little, the iteration no longer converges. In particular, the initial guess for the chimney length given by equation (5.9), which isL⁽⁰⁾_C = 204.49 mm in this case, is out of the region of conver-gence. Apparently, as the initial chimney length is increased, the residual abruptly increases after a certain point; whereas if the initial chimney length is decreased, the residual increases in a gradual manner in a narrow region. When convergence is not reached the algorithm gets stuck around a suboptimal solution and the residual ceases to decrease after a few iterations.

The heuristic iterative method introduced in Section 5.2.2 provides convergence in the whole region shown in Figure 5.5 with a maximum of eight iterations and tends to the same optimal re-sult in the whole region. Since this iterative method does not require the evaluation of sensitivity functions, it is much more efficient computationally. From these results, it can be assessed that the performance of the sensitivity approach in its current form is inferior compared to the heuristic iterative algorithm for chimney pipe length optimization. It can be argued that the sensitivity-based approach can be improved or optimized for the special case of chimney pipe optimization;

however, such examinations are out of the scope of the study presented in this chapter.